Re: Three questions about statistical mechanics [Archive]

Erik

Apr7-04, 09:08 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Arnold Neumaier wrote:\n> Erik wrote:\n> >>\n> >>>QUESTION 1: [...] But what is the quantum version of the\n> >>>Boltzmann entropy analogue?\n> >>\n> >>There are quantum Boltzmann equations, which answer that.\n> >\n> > Isn\'t the quantum Boltzmann equation a dynamical equation for the\n> > density matrix ? I\'m looking for the quantum analogue of the logarithm\n> > of the phase space volume, i.e. I\'m just looking for a definition.\n>\n> Maybe your question was ambiguous. I understood Boltzmann entropy\n> as the entropy in the Botzmann equation that increases with time.\n> This has a quantum analogue.\n>\n> But you seem to ask for the quantum analogue of the microcanonical\n> ensemble, it seems now. (I don\'t understand your question,\n> so I am just guessing.) Look into statistical mechanics book.\n> Many books treat the classical and the quantum case side by side.\n> And by comparing, you can find out what corresponds to each other.\n\nI am looking for the _definition_ of quantum Boltzmann entropy\nanalogue, not a dynamical equation that gives its time evolution in a\nrestricted case. As I was looking for a suitable reference for a point\nI\'ll mention below, I actually stumbled across the definition. The\nquantum Boltzmann entropy analogue of a state vector psi is\n\nSB(psi) = \\sum_a p_a(psi) log(dim(G_a)) - \\sum_a p_a(psi)\nlog(p_a(psi)),\n\nwhere G_a is a subspace of the total Hilbert space corresponding to\nthe situation that a set of commuting observables have have values in\nsome range that define macrostate a, and p_a is the scalar product of\npsi with the projection of psi into G_a. This definition is given as\neq. (3) in\n\nLebowitz J.L. (1999) "Microscopic Origins of Irreversible Macroscopic\nBehavior", Physica A, 263:516-527\nhttp://dx.doi.org/10.1016/S0378-4371(98)00514-7\n\n> >>>QUESTION 3: Is there a good textbook or review article that discusses\n> >>>the entropy analogues with attention to their different properties and\n> >>>implications? Most textbooks on thermodynamics are extremely unhelpful\n> >>>when it comes to comparing different entropy analogues.\n> >>\n> >>Entropy analogues are outside mainstream physics and hence not\n> >>represented in textbooks, except sometimes in passing.\n> >>They are not very useful, either.\n>\n> > But don\'t they form the basis of equilibrium statistical mechanics?\n>\n> This has as basis _the_ entropy, not entropy analogues like\n> Tsallis entropy. Entropy is a very well established concept,\n> and tinkering with it is a sign of fringe physics.\n\nBut there are two at least two competing camps in statistical\nmechanics -- the Boltzmann camp and the Gibbs camp. The classical\nBoltzmann entropy analogue is a function of a single phase space point\nand the classical Gibbs entropy analogue is a function of a\nprobability density on the phase space. The classical Boltzmann\nentropy analogue makes sense without probabilities, the classical\nGibbs entropy analogue presupposes probabilities. The classical\nBoltzmann entropy can both increase and decrease in isolated systems.\nThe classical Gibbs entropy analogue is a conserved quantity in\nisolated systems.\n\nThe Boltzmann camp thinks that the Boltzmann entropy analogue is _the_\nentropy and the Gibbs camp thinks that the Gibbs entropy analogue is\n_the_ entropy. To give just two examples: Lebowitz belong to the\nBoltzmann camp, Jaynes belonged to the Gibbs camp.\n\nIt seems to me that entropy is actually not a very well-established\nconcept when we are at the "foundations" level. Different prominent\nphysicists say different, inequivalent things about it and, unlike\nquantum mechanics were everyone agrees on the basic mathematical\nformalism, it is not just as matter of philosophical interpretations.\nAlthough my problems are not restricted to the statistical mechanical\nversion of the second law of thermodynamics, I can\'t resist quoting\nBricmont:\n\n"The Second Law seems now a bit difficult to state precisely. \'Entropy\nincreases\'; yes, but which one?"\n\nfrom p. 28 of http://www.arxiv.org/abs/chao-dyn/9603009\n\n> > Most authors base their texts on one entropy analogue, but surely\n> > someone has discussed them all? I can\'t be the first person to become\n> > suspicious by the fact that there at least two inequivalent versions\n> > of statistical mechanics.\n>\n> There is standard statistical mechanics that explains thermodynamics,\n> and there may be variants that are of little relevance, though they\n> can be studied for the sake of curiosity or for some special\n> applications not directly related to the standard applications.\n\nI\'m not referring to attempts to develop new statistical mechanics for\nnew problem domains by changing the Gibbs entropy analogue to, say,\nthe Tsallis entropy. I am referring to facts like that despite having\nworked on it for more than a century physicists still haven\'t reached\na consensus on questions like: Is the entropy of statistical mechanics\na function of a single phase space point or a functional of a\nprobability density on the phase space or something else?\n\nSuch lack of consensus is normal, but it is confusing for a student\nlike me to have to _discover_ it, rather being warned about it by\nlecturers and textbooks. I was taught that the Boltzmann entropy is a\nspecial case of the Gibbs entropy, when they are in fact entirely\ndifferent concepts.\n\n> > Studying statistical mechanics has apparently made me so confused that\n> > I can\'t even count anymore! There was a time when I thought I actually\n> > understood thermodynamics. Then I discovered that there are at least\n> > two inequivalent approaches to statistical mechanics (three if we\n> > count Jaynes\'s MaxEnt Bayesian approach as different from the\n> > traditional Gibbs approach). And to make it worse I also read Uffink\'s\n> > "Bluff your way in the second law of thermodynamics" and now I don\'t\n> > even know what the second law says.\n>\n> He was bluffing you...\n>\n>\n> > It feels like my confusion\n> > increases irreversibly the more I look at the foundations of\n> > statistical mechanics.\n>\n> This is an intermediate stage only. Remember the hero in a fairy-tale?\n> Persistence is the key to victory!\n>\n> Well, people don\'t agree on what exactly the second law is.\n> The various approaches to statistical mechanics are equivalent as\n> far as the final formulas with which one calculates real properties\n> are concerned. Everything else is just motivating blabla, and\n> different people find different things motivating.\n\nThat seems like a useful pragmatic approach one can fall back on when\nthings get tough.\n\n> Apropos foundations, I can recommend the book\n> L. Sklar,\n> Physics and Chance,\n> Cambridge Univ. Press, Cambridge 1993.\n> Treats only the classical side of it, but does this more thorough\n> than anyone else.\n>\n> If you want to know my personal view on the matter,\n> write me an email offline, and I\'ll send you a draft of a paper\n> on the foundations of thermodynamics.\n\nThanks. An offline e-mail is on its way.\n\n> I found it very valuable in learning about foundations to take for\n> really \'real\' only the stuff with which people calculate experimentally\n> relevant results, and to consider everything else as sort of\n> handwaving background of more or less relevance, but never to\n> be taken too seriously. This helps a lot to remain sane in the\n> cloudy worlds of the foundations...\n\nDo you regard the entropy of a system (or _changes_ in the entropy of\na system) as experimentally relevant? It would simplify matters if I\ncould regard the entropy in statistical mechanics as something\nunmeasurable whose only function is to generate predictions about,\nsay, pressures, magnetization, etc. But this presupposes that entropy\n(and changes in entropy) can\'t be measured.\n\nErik\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Arnold Neumaier wrote:
> Erik wrote:
> >>
> >>>QUESTION 1: [...] But what is the quantum version of the
> >>>Boltzmann entropy analogue?
> >>
> >>There are quantum Boltzmann equations, which answer that.
> >
> > Isn't the quantum Boltzmann equation a dynamical equation for the
> > density matrix ? I'm looking for the quantum analogue of the logarithm
> > of the phase space volume, i.e. I'm just looking for a definition.
>
> Maybe your question was ambiguous. I understood Boltzmann entropy
> as the entropy in the Botzmann equation that increases with time.
> This has a quantum analogue.
>
> But you seem to ask for the quantum analogue of the microcanonical
> ensemble, it seems now. (I don't understand your question,
> so I am just guessing.) Look into statistical mechanics book.
> Many books treat the classical and the quantum case side by side.
> And by comparing, you can find out what corresponds to each other.

I am looking for the _definition_ of quantum Boltzmann entropy
analogue, not a dynamical equation that gives its time evolution in a
restricted case. As I was looking for a suitable reference for a point
I'll mention below, I actually stumbled across the definition. The
quantum Boltzmann entropy analogue of a state vector \psi is

SB(\psi) = \sum_a p_a(\psi) log(dim(G_a)) - \sum_a p_a(\psi)[/itex]
[itex]log(p_a(\psi)),

where G_a is a subspace of the total Hilbert space corresponding to
the situation that a set of commuting observables have have values in
some range that define macrostate a, and p_a is the scalar product of
\psi with the projection of \psi into G_a. This definition is given as
eq. (3) in

Lebowitz J.L. (1999) "Microscopic Origins of Irreversible Macroscopic
Behavior", Physica A, 263:516-527
http://dx.doi.org/10.1016/S0378-4371(98)00514-7

> >>>QUESTION 3: Is there a good textbook or review article that discusses
> >>>the entropy analogues with attention to their different properties and
> >>>implications? Most textbooks on thermodynamics are extremely unhelpful
> >>>when it comes to comparing different entropy analogues.
> >>
> >>Entropy analogues are outside mainstream physics and hence not
> >>represented in textbooks, except sometimes in passing.
> >>They are not very useful, either.
>
> > But don't they form the basis of equilibrium statistical mechanics?
>
> This has as basis _the_ entropy, not entropy analogues like
> Tsallis entropy. Entropy is a very well established concept,
> and tinkering with it is a sign of fringe physics.

But there are two at least two competing camps in statistical
mechanics -- the Boltzmann camp and the Gibbs camp. The classical
Boltzmann entropy analogue is a function of a single phase space point
and the classical Gibbs entropy analogue is a function of a
probability density on the phase space. The classical Boltzmann
entropy analogue makes sense without probabilities, the classical
Gibbs entropy analogue presupposes probabilities. The classical
Boltzmann entropy can both increase and decrease in isolated systems.
The classical Gibbs entropy analogue is a conserved quantity in
isolated systems.

The Boltzmann camp thinks that the Boltzmann entropy analogue is _the_
entropy and the Gibbs camp thinks that the Gibbs entropy analogue is
_the_ entropy. To give just two examples: Lebowitz belong to the
Boltzmann camp, Jaynes belonged to the Gibbs camp.

It seems to me that entropy is actually not a very well-established
concept when we are at the "foundations" level. Different prominent
physicists say different, inequivalent things about it and, unlike
quantum mechanics were everyone agrees on the basic mathematical
formalism, it is not just as matter of philosophical interpretations.
Although my problems are not restricted to the statistical mechanical
version of the second law of thermodynamics, I can't resist quoting
Bricmont:

"The Second Law seems now a bit difficult to state precisely. 'Entropy
increases'; yes, but which one?"

from p. 28 of http://www.arxiv.org/abs/chao-dyn/9603009

> > Most authors base their texts on one entropy analogue, but surely
> > someone has discussed them all? I can't be the first person to become
> > suspicious by the fact that there at least two inequivalent versions
> > of statistical mechanics.
>
> There is standard statistical mechanics that explains thermodynamics,
> and there may be variants that are of little relevance, though they
> can be studied for the sake of curiosity or for some special
> applications not directly related to the standard applications.

I'm not referring to attempts to develop new statistical mechanics for
new problem domains by changing the Gibbs entropy analogue to, say,
the Tsallis entropy. I am referring to facts like that despite having
worked on it for more than a century physicists still haven't reached
a consensus on questions like: Is the entropy of statistical mechanics
a function of a single phase space point or a functional of a
probability density on the phase space or something else?

Such lack of consensus is normal, but it is confusing for a student
like me to have to _discover_ it, rather being warned about it by
lecturers and textbooks. I was taught that the Boltzmann entropy is a
special case of the Gibbs entropy, when they are in fact entirely
different concepts.

> > Studying statistical mechanics has apparently made me so confused that
> > I can't even count anymore! There was a time when I thought I actually
> > understood thermodynamics. Then I discovered that there are at least
> > two inequivalent approaches to statistical mechanics (three if we
> > count Jaynes's MaxEnt Bayesian approach as different from the
> > traditional Gibbs approach). And to make it worse I also read Uffink's
> > "Bluff your way in the second law of thermodynamics" and now I don't
> > even know what the second law says.
>
> He was bluffing you...
>
>
> > It feels like my confusion
> > increases irreversibly the more I look at the foundations of
> > statistical mechanics.
>
> This is an intermediate stage only. Remember the hero in a fairy-tale?
> Persistence is the key to victory!
>
> Well, people don't agree on what exactly the second law is.
> The various approaches to statistical mechanics are equivalent as
> far as the final formulas with which one calculates real properties
> are concerned. Everything else is just motivating blabla, and
> different people find different things motivating.

That seems like a useful pragmatic approach one can fall back on when
things get tough.

> Apropos foundations, I can recommend the book
> L. Sklar,
> Physics and Chance,
> Cambridge Univ. Press, Cambridge 1993.
> Treats only the classical side of it, but does this more thorough
> than anyone else.
>
> If you want to know my personal view on the matter,
> write me an email offline, and I'll send you a draft of a paper
> on the foundations of thermodynamics.

Thanks. An offline e-mail is on its way.

> I found it very valuable in learning about foundations to take for
> really 'real' only the stuff with which people calculate experimentally
> relevant results, and to consider everything else as sort of
> handwaving background of more or less relevance, but never to
> be taken too seriously. This helps a lot to remain sane in the
> cloudy worlds of the foundations...

Do you regard the entropy of a system (or _changes_ in the entropy of
a system) as experimentally relevant? It would simplify matters if I
could regard the entropy in statistical mechanics as something
unmeasurable whose only function is to generate predictions about,
say, pressures, magnetization, etc. But this presupposes that entropy
(and changes in entropy) can't be measured.

Erik

Arnold Neumaier

Apr8-04, 02:26 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Erik wrote:\n>\n> I am looking for the _definition_ of quantum Boltzmann entropy\n> analogue, not a dynamical equation that gives its time evolution in a\n> restricted case. As I was looking for a suitable reference for a point\n> I\'ll mention below, I actually stumbled across the definition. The\n> quantum Boltzmann entropy analogue of a state vector psi is\n>\n> SB(psi) = \\sum_a p_a(psi) log(dim(G_a)) - \\sum_a p_a(psi)\n> log(p_a(psi)),\n>\n> where G_a is a subspace of the total Hilbert space corresponding to\n> the situation that a set of commuting observables have have values in\n> some range that define macrostate a, and p_a is the scalar product of\n> psi with the projection of psi into G_a.\n\nIn the special case where all dim G_a =d,\nand the formula becomes (up to a constant shift d)\nSB = - \\sum_a p_a log(p_a),\nThis shows that the Boltzmann entropy is a\ncoarse-grained, approximate version\nS_b = - trace rho_b log rho_b\nof the Gibbs entropy\nS_g = - trace rho_g log rho_g.\nHere rho_g is the density matrix of the microscopic system,\nand rho_b is the coarse-grained density of its macroscopic view.\nIn a basis of eigenstates of the set of relevant commuting observables,\nrho_b is a diagonal matrix withe entries\np_a = trace rho_b P_a,\nwhere the P_a are commuting projectors adding up to 1.\n\n\n> But there are two at least two competing camps in statistical\n> mechanics -- the Boltzmann camp and the Gibbs camp.\n\nThis is the same camp, with Gibbs giving the microscopic view\nand Boltzmann the coarse-grained view.\n\n\n> It seems to me that entropy is actually not a very well-established\n> concept when we are at the "foundations" level. Different prominent\n> physicists say different, inequivalent things about it\n\nThey talk about different levels of the same thing.\n\n\n> from p. 28 of http://www.arxiv.org/abs/chao-dyn/9603009\n\nOn p.27, this author explains the contextual nature of Boltzmann\nentropies: one exists for each choice of coarse graining.\nThe finest coarse graining possible (a maximal commuting set\nof observables) is the only objective one among these, and this is\nprecisely the Gibbs entropy.\n\n\n> I\'m not referring to attempts to develop new statistical mechanics for\n> new problem domains by changing the Gibbs entropy analogue to, say,\n> the Tsallis entropy. I am referring to facts like that despite having\n> worked on it for more than a century physicists still haven\'t reached\n> a consensus on questions like: Is the entropy of statistical mechanics\n> a function of a single phase space point or a functional of a\n> probability density on the phase space or something else?\n\nIt depends on the context, that\'s all.\n\n\n> Do you regard the entropy of a system (or _changes_ in the entropy of\n> a system) as experimentally relevant? It would simplify matters if I\n> could regard the entropy in statistical mechanics as something\n> unmeasurable whose only function is to generate predictions about,\n> say, pressures, magnetization, etc. But this presupposes that entropy\n> (and changes in entropy) can\'t be measured.\n\nNo. Entropy can be measured (though somewhat indirectly, as many\nother quantities of physical interest). It is experimentally very\nrelevant, as can be seen by its prominent role in thermodynamics\n(already without statistical mechanics). For example, the fact that\na physical system is isolated (mechanically, thermally, and chemically)\nis reflected in the time-independence of the entropy, while entropy\nincreases in systems coupled to degrees of freedom unobservable\nat the level of modeling used.\n\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Erik wrote:
>
> I am looking for the _definition_ of quantum Boltzmann entropy
> analogue, not a dynamical equation that gives its time evolution in a
> restricted case. As I was looking for a suitable reference for a point
> I'll mention below, I actually stumbled across the definition. The
> quantum Boltzmann entropy analogue of a state vector \psi is
>
> SB(\psi) = \sum_a p_a(\psi) log(dim(G_a)) - \sum_a p_a(\psi)
> log(p_a(\psi)),
>
> where G_a is a subspace of the total Hilbert space corresponding to
> the situation that a set of commuting observables have have values in
> some range that define macrostate a, and p_a is the scalar product of
> \psi with the projection of \psi into G_a.

In the special case where all dim G_a =d,
and the formula becomes (up to a constant shift d)
SB = - \sum_a p_a log(p_a),
This shows that the Boltzmann entropy is a
coarse-grained, approximate version
S_b = - trace \rho_b log \rho_b
of the Gibbs entropy
S_g = - trace \rho_g log \rho_g.
Here \rho_g is the density matrix of the microscopic system,
and \rho_b is the coarse-grained density of its macroscopic view.
In a basis of eigenstates of the set of relevant commuting observables,
\rho_b is a diagonal matrix withe entries
p_a = trace \rho_b P_a,
where the P_a are commuting projectors adding up to 1.

> But there are two at least two competing camps in statistical
> mechanics -- the Boltzmann camp and the Gibbs camp.

This is the same camp, with Gibbs giving the microscopic view
and Boltzmann the coarse-grained view.

> It seems to me that entropy is actually not a very well-established
> concept when we are at the "foundations" level. Different prominent
> physicists say different, inequivalent things about it

They talk about different levels of the same thing.

> from p. 28 of http://www.arxiv.org/abs/chao-dyn/9603009

On p.27, this author explains the contextual nature of Boltzmann
entropies: one exists for each choice of coarse graining.
The finest coarse graining possible (a maximal commuting set
of observables) is the only objective one among these, and this is
precisely the Gibbs entropy.

> I'm not referring to attempts to develop new statistical mechanics for
> new problem domains by changing the Gibbs entropy analogue to, say,
> the Tsallis entropy. I am referring to facts like that despite having
> worked on it for more than a century physicists still haven't reached
> a consensus on questions like: Is the entropy of statistical mechanics
> a function of a single phase space point or a functional of a
> probability density on the phase space or something else?

It depends on the context, that's all.

> Do you regard the entropy of a system (or _changes_ in the entropy of
> a system) as experimentally relevant? It would simplify matters if I
> could regard the entropy in statistical mechanics as something
> unmeasurable whose only function is to generate predictions about,
> say, pressures, magnetization, etc. But this presupposes that entropy
> (and changes in entropy) can't be measured.

No. Entropy can be measured (though somewhat indirectly, as many
other quantities of physical interest). It is experimentally very
relevant, as can be seen by its prominent role in thermodynamics
(already without statistical mechanics). For example, the fact that
a physical system is isolated (mechanically, thermally, and chemically)
is reflected in the time-independence of the entropy, while entropy
increases in systems coupled to degrees of freedom unobservable
at the level of modeling used.

Arnold Neumaier

Erik

Apr22-04, 03:34 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<40745860.3040607@univie.ac.at>...\n> Erik wrote:\n> >\n> > I am looking for the _definition_ of quantum Boltzmann entropy\n> > analogue, not a dynamical equation that gives its time evolution in a\n> > restricted case. As I was looking for a suitable reference for a point\n> > I\'ll mention below, I actually stumbled across the definition. The\n> > quantum Boltzmann entropy analogue of a state vector psi is\n> >\n> > SB(psi) = \\sum_a p_a(psi) log(dim(G_a)) - \\sum_a p_a(psi)\n> > log(p_a(psi)),\n> >\n> > where G_a is a subspace of the total Hilbert space corresponding to\n> > the situation that a set of commuting observables have have values in\n> > some range that define macrostate a, and p_a is the scalar product of\n> > psi with the projection of psi into G_a.\n>\n> In the special case where all dim G_a =d,\n> and the formula becomes (up to a constant shift d)\n> SB = - \\sum_a p_a log(p_a),\n> This shows that the Boltzmann entropy is a\n> coarse-grained, approximate version\n> S_b = - trace rho_b log rho_b\n> of the Gibbs entropy\n> S_g = - trace rho_g log rho_g.\n> Here rho_g is the density matrix of the microscopic system,\n> and rho_b is the coarse-grained density of its macroscopic view.\n> In a basis of eigenstates of the set of relevant commuting observables,\n> rho_b is a diagonal matrix withe entries\n> p_a = trace rho_b P_a,\n> where the P_a are commuting projectors adding up to 1.\n\nI don\'t quite follow this. The quantity S_g is always zero for a pure\nstate. Only for mixed states can S_g be non-zero. The quantum\nBoltzmann entropy, on the other hand, can be non-zero for pure states.\nIt is therefore not possible that the quantum Boltzmann entropy is a\ncoarse-grained version of S_g.\n\n> > But there are two at least two competing camps in statistical\n> > mechanics -- the Boltzmann camp and the Gibbs camp.\n>\n> This is the same camp, with Gibbs giving the microscopic view\n> and Boltzmann the coarse-grained view.\n\nWhat you call the "microscopic view" and the "coarse-grained view" are\nvery different at a foundational level. For instance, the Gibbs\nentropy is conserved in an isolated system, whereas the Boltzmann\nentropy is not. I don\'t think it is possible for a single camp to\nsustain both these views at the same time.\n\n> > It seems to me that entropy is actually not a very well-established\n> > concept when we are at the "foundations" level. Different prominent\n> > physicists say different, inequivalent things about it\n>\n> They talk about different levels of the same thing.\n>\n>\n> > from p. 28 of http://www.arxiv.org/abs/chao-dyn/9603009\n>\n> On p.27, this author explains the contextual nature of Boltzmann\n> entropies: one exists for each choice of coarse graining.\n> The finest coarse graining possible (a maximal commuting set\n> of observables) is the only objective one among these, and this is\n> precisely the Gibbs entropy.\n\nYou cannot get from the Boltzmann entropy (which is defined in terms\nof phase space volume and _not_ probabilities) to the Gibbs entropy\n(which is defined in terms of probabilities) by "fine-graining".\n\nGiven a partitioning of a phase space, the Boltzmann entropy of the\nphase space point x is defined as\n\nSB(x) = k log |M(x)|\n\nwhere M(x) is the phase space partition to which x belongs. Physicists\nfrom the Boltzmann camp refer to M(x) as the "macrostate".\n\nThe Gibbs entropy requires either a partition on the phase space as\nwell as a discrete probability distribution over this partition, or a\ncontinuous probability density on the phase space. Physicists from the\nGibbs camp refer to this probability distribution/density as the\n"macrostate".\n\nThe path from the Boltzmann entropy to the Gibbs entropy is to turn\nattention away from single phase space points to probability densities\nover phase space points. It is not a matter of fine- or\ncoarse-graining.\n\n> > I\'m not referring to attempts to develop new statistical mechanics for\n> > new problem domains by changing the Gibbs entropy analogue to, say,\n> > the Tsallis entropy. I am referring to facts like that despite having\n> > worked on it for more than a century physicists still haven\'t reached\n> > a consensus on questions like: Is the entropy of statistical mechanics\n> > a function of a single phase space point or a functional of a\n> > probability density on the phase space or something else?\n>\n> It depends on the context, that\'s all.\n\nIf you mean that the Boltzmann entropy depends on how we partition the\nphase space, I agree. If you mean that the choice between Boltzmann\nentropy and Gibbs entropy depends on the context, I disagree. No other\nquantity in physics that I know of has this kind of split personality.\nEnergy is always a function of a single phase space point. Momentum\nalways is a phase space coordinate. Etc.\n\nBut statical mechanical entropy, unlike e.g. energy and momentum, has\na split personality in the sense that some physicists define it as a\nfunction of a single phase space point while other physicists define\nit as a function(al) of a probability distribution/density.\n\nRegards,\nErik\n\n> > Do you regard the entropy of a system (or _changes_ in the entropy of\n> > a system) as experimentally relevant? It would simplify matters if I\n> > could regard the entropy in statistical mechanics as something\n> > unmeasurable whose only function is to generate predictions about,\n> > say, pressures, magnetization, etc. But this presupposes that entropy\n> > (and changes in entropy) can\'t be measured.\n>\n> No. Entropy can be measured (though somewhat indirectly, as many\n> other quantities of physical interest). It is experimentally very\n> relevant, as can be seen by its prominent role in thermodynamics\n> (already without statistical mechanics). For example, the fact that\n> a physical system is isolated (mechanically, thermally, and chemically)\n> is reflected in the time-independence of the entropy, while entropy\n> increases in systems coupled to degrees of freedom unobservable\n> at the level of modeling used.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<40745860.3040607@univie.ac.at>...
> Erik wrote:
> >
> > I am looking for the _definition_ of quantum Boltzmann entropy
> > analogue, not a dynamical equation that gives its time evolution in a
> > restricted case. As I was looking for a suitable reference for a point
> > I'll mention below, I actually stumbled across the definition. The
> > quantum Boltzmann entropy analogue of a state vector \psi is
> >
> > SB(\psi) = \sum_a p_a(\psi) log(dim(G_a)) - \sum_a p_a(\psi)
> > log(p_a(\psi)),
> >
> > where G_a is a subspace of the total Hilbert space corresponding to
> > the situation that a set of commuting observables have have values in
> > some range that define macrostate a, and p_a is the scalar product of
> > \psi with the projection of \psi into G_a.
>
> In the special case where all dim G_a =d,
> and the formula becomes (up to a constant shift d)
> SB = - \sum_a p_a log(p_a),
> This shows that the Boltzmann entropy is a
> coarse-grained, approximate version
> S_b = - trace \rho_b log \rho_b
> of the Gibbs entropy
> S_g = - trace \rho_g log \rho_g.
> Here \rho_g is the density matrix of the microscopic system,
> and \rho_b is the coarse-grained density of its macroscopic view.
> In a basis of eigenstates of the set of relevant commuting observables,
> \rho_b is a diagonal matrix withe entries
> p_a = trace \rho_b P_a,
> where the P_a are commuting projectors adding up to 1.

I don't quite follow this. The quantity S_g is always zero for a pure
state. Only for mixed states can S_g be non-zero. The quantum
Boltzmann entropy, on the other hand, can be non-zero for pure states.
It is therefore not possible that the quantum Boltzmann entropy is a
coarse-grained version of S_g.

> > But there are two at least two competing camps in statistical
> > mechanics -- the Boltzmann camp and the Gibbs camp.
>
> This is the same camp, with Gibbs giving the microscopic view
> and Boltzmann the coarse-grained view.

What you call the "microscopic view" and the "coarse-grained view" are
very different at a foundational level. For instance, the Gibbs
entropy is conserved in an isolated system, whereas the Boltzmann
entropy is not. I don't think it is possible for a single camp to
sustain both these views at the same time.

> > It seems to me that entropy is actually not a very well-established
> > concept when we are at the "foundations" level. Different prominent
> > physicists say different, inequivalent things about it
>
> They talk about different levels of the same thing.
>
>
> > from p. 28 of http://www.arxiv.org/abs/chao-dyn/9603009
>
> On p.27, this author explains the contextual nature of Boltzmann
> entropies: one exists for each choice of coarse graining.
> The finest coarse graining possible (a maximal commuting set
> of observables) is the only objective one among these, and this is
> precisely the Gibbs entropy.

You cannot get from the Boltzmann entropy (which is defined in terms
of phase space volume and _not_ probabilities) to the Gibbs entropy
(which is defined in terms of probabilities) by "fine-graining".

Given a partitioning of a phase space, the Boltzmann entropy of the
phase space point x is defined as

SB(x) = k[/itex] log [itex]|M(x)|

where M(x) is the phase space partition to which x belongs. Physicists
from the Boltzmann camp refer to M(x) as the "macrostate".

The Gibbs entropy requires either a partition on the phase space as
well as a discrete probability distribution over this partition, or a
continuous probability density on the phase space. Physicists from the
Gibbs camp refer to this probability distribution/density as the
"macrostate".

The path from the Boltzmann entropy to the Gibbs entropy is to turn
attention away from single phase space points to probability densities
over phase space points. It is not a matter of fine- or
coarse-graining.

> > I'm not referring to attempts to develop new statistical mechanics for
> > new problem domains by changing the Gibbs entropy analogue to, say,
> > the Tsallis entropy. I am referring to facts like that despite having
> > worked on it for more than a century physicists still haven't reached
> > a consensus on questions like: Is the entropy of statistical mechanics
> > a function of a single phase space point or a functional of a
> > probability density on the phase space or something else?
>
> It depends on the context, that's all.

If you mean that the Boltzmann entropy depends on how we partition the
phase space, I agree. If you mean that the choice between Boltzmann
entropy and Gibbs entropy depends on the context, I disagree. No other
quantity in physics that I know of has this kind of split personality.
Energy is always a function of a single phase space point. Momentum
always is a phase space coordinate. Etc.

But statical mechanical entropy, unlike e.g. energy and momentum, has
a split personality in the sense that some physicists define it as a
function of a single phase space point while other physicists define
it as a function(al) of a probability distribution/density.

Regards,
Erik

> > Do you regard the entropy of a system (or _changes_ in the entropy of
> > a system) as experimentally relevant? It would simplify matters if I
> > could regard the entropy in statistical mechanics as something
> > unmeasurable whose only function is to generate predictions about,
> > say, pressures, magnetization, etc. But this presupposes that entropy
> > (and changes in entropy) can't be measured.
>
> No. Entropy can be measured (though somewhat indirectly, as many
> other quantities of physical interest). It is experimentally very
> relevant, as can be seen by its prominent role in thermodynamics
> (already without statistical mechanics). For example, the fact that
> a physical system is isolated (mechanically, thermally, and chemically)
> is reflected in the time-independence of the entropy, while entropy
> increases in systems coupled to degrees of freedom unobservable
> at the level of modeling used.